On harmonic and subharmonic solutions of nonlinear second order equations: Symmetry and bifurcation

Abstract

Consider the equation ü + u = g(u,p) + μf(t), where p, μ are small parameters, f is an even continuous п/m-odd-harmonic function (i.e., f(t+п/m) = -f(t), for every t in R), m≥2 and g is an odd function of u. Under certain conditions on f and g it is proved that the small 2п-periodic solutions of the above equation maintain some symmetry properties of the forcing f(t), when μ ≠ 0. Other interesting results describe the changes of the number of such solutions, as p and μ vary in a small neighborhood of the origin. As another contribution of this paper, it was proved that a central assumption which was required in the main results, is generic. The main tool used in this work is the Liapunov-Schmidt Method. © 1990, Taylor & Francis Group, LLC. All rights reserved.